Mathematics previous year questions Of Statistics For NDA

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previous year questions Of Statistics For NDA

previous year questions Of Statistics

Set - 1

Q 2713791649

The variance of 20 observations is 5. If each observation is multiplied by 3,
then what is the new variance of the resulting observations?
NDA Paper 1 2017

(A)

`5`

(B)

`10`

(C)

`15`

(D)

`45`

Solution:

If each observation is multiplied by 3 the variance will become `3^2` times the original value as per variance formula.

Therefore the new variance will be 45.

Correct Answer is `=>` (D) `45`

Q 2723891741

The mean of a group of `100` observations was found to be `20`. Later it was found that four observations were ·incorrect, which were recorded as `21, 21, 18` and `20`. What is the mean if the incorrect observations are omitted?
NDA Paper 1 2017

(A)

`18`

(B)

`20`

(C)

`21`

(D)

`22`

Solution:

As the mean of incorrect observations is 20, removing these observations will not effect average.

Correct Answer is `=>` (B) `20`

Q 2763891745

The mean weight of 150 students in a certain class is 60 kg. The mean weight of boys in the class is 70 kg and that of
girls is 55 kg. What is the number of boys in the class?
NDA Paper 1 2017

(A)

`50`

(B)

`55`

(C)

`60`

(D)

`100`

Solution:

Let no of boy is b & no. of girls is g , `barz = 60, b+g = 150`

`barb = 70, ` `barg = 55`

Total weight

`150*60 = b*70 + (150-b)55`

`b= 50`

Alternatively :

let No. of Boys `=n`

No of girls `= 150-n`

`(sum_(i=1)^n x_i + sum_(n-150)^n x_i)/(150) = 60`.................(i)

`(sum_(i=1)^n x_i)/n = 70`........................(ii)

`(sum_(n-150)^150 x_i)/(150-n) = 55`...............(iii)

by (i), (ii), & (iii)

`70 n+ 55 (150-n) = 60 xx 150`

`n=50`

Correct Answer is `=>` (A) `50`

Q 2733091842

In an examination, `40 %` of candidates
got second class. When the data are
represented by a pie chart, what is the
angle· corresponding to second class?
NDA Paper 1 2017

(A)

`40^o`

(B)

`90^o`

(C)

`144 ^o`

(D)

`320 ^o`

Solution:

`40 %` of candidates got second class.

then angel will be `40 %` of `360^o`

`360^o xx 40/100`

`=144^o`

Correct Answer is `=>` (C) `144 ^o`

Q 2753091844

Consider· the following statements
Statement 1 : Range is not a good measure of dispersion.

Statement 2 : Range is highly affected by the existence of extreme values.

Which one of the following is correct in respect of the above statements?
NDA Paper 1 2017

(A)

Both Statement I and Statement 2 are correct and Statement 2 is the correct explanation of Statement 1

(B)

Both State~erit I and Statement 2 are correct but Statement 2 is not the· correct explanation of Statement 1

(C)

Statement 1 is correct but Statement 2 is not correct

(D)

Statement 2 is correct but Statement 1 is not correct

Solution:

This is obvious. Statement 2 explains the assertion. This question can be done by anyone without even knowing the subject.

Correct Answer is `=>` (A) Both Statement I and Statement 2 are correct and Statement 2 is the correct explanation of Statement 1

Q 2703091848

If the data are moderately non-symmetrical, then which one of the following empirical relationships is
correct?
NDA Paper 1 2017

(A)

2 x Standard deviation= 5 x Mean deviation

(B)

5 x Standard deviation = 2 x Mean deviation

(C)

4 x Standard deviation = 5 x Mean deviation

(D)

5 x Standard deviation = 4 x Mean deviation

Solution:

we know that, If the data are moderately non-symmetrical, following relationships hold :

4 S.D = 5 M.D = 6 Q.D

here Q.D. = Quartile Deviation

Correct Answer is `=>` (C) 4 x Standard deviation = 5 x Mean deviation

Q 2713191940

Data can be represented in which of the following forms?

1. Textual form
2. ·Tabular form
3. Graphical form

Select the correct-answer using the code given below ..
NDA Paper 1 2017

(A)

1 and 2 only

(B)

2 and 3 only

(C)

1 and 3 only

(D)

1, 2 and 3

Solution:

Well, it is obvious

Correct Answer is `=>` (D) 1, 2 and 3

Q 2723191941

For given statistical data, the graphs for less than ogive and more' than ogive are "drawn. If the point at which the two curves intersect is P, then abscissa of point . P gives the value of which one of the following measures of central tendency?
NDA Paper 1 2017

(A)

Median

(B)

Mean

(C)

Mode

(D)

Geometric mean

Solution:

A curve that represents the cumulative frequency distribution of grouped data is called an ogive or cumulative frequency curve.

The two types of Ogives are more than type ogive and less than type ogive. An ogive representing a cumulative frequency distribution of ‘more than’ type is called a more than ogive. An ogive representing a cumulative frequency distribution of ‘less than’ type is called a less than ogive.

Ogives can be used to find the median of a grouped data. The median of grouped data can be obtained graphically by plotting the Ogives of the less than type and more than type and locate the point of intersection of both the Ogives. The x-coordinate of the point of intersection of two Ogives gives the median of the grouped data

Correct Answer is `=>` (A) Median

Q 2753191944

If the regression coefficient of x on y and y on x are `-1/2` and `-1/8` respectively, then what is the correlation coefficient
between x and y?
NDA Paper 1 2017

(A)

`-1/4`

(B)

`-1/16`

(C)

`1/16`

(D)

`1/4`

Solution:

`b_(yx) =(-1)/2`

`b_(xy) =(-1)/8`

`r= sqrt(b_(yx) * b_(xy)`

`= sqrt((-1/2) (-1/8))`

`=1/4`

Correct Answer is `=>` (A) `-1/4`

Q 2703191948

A sample of 5 observations has mean 32 and median 33. Later it is.found that an observation was recorded incorrectly as 40 instead of 35. If we correct the data, then which one of the . following is correct?
NDA Paper 1 2017

(A)

The mean and median remain the same

(B)

The median remains the same but the mean will decrease

(C)

The mean and median both will decrease

(D)

The mean remains ·the same but median will decrease

Solution:

Let the numbers be

20 , 32 , 33, 35, 40

If we replace 40 by 35 then

median will remain same

and mean will become = ` (20+32+33+35+35) / 5 `

= 31

`=>` mean will decrease

Correct Answer is `=>` (B) The median remains the same but the mean will decrease

Q 2724145051

The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is· the number or'
trials?
NDA Paper 1 2017

(A)

`2`

(B)

`12`

(C)

`18`

(D)

`24`

Solution:

`np = 12`

`npq = 2^2 = 4`

` q = 1/3 => p = 1/3 `

`=> n = 18`

Correct Answer is `=>` (C) `18`

Q 2137180082

What is the mean deviation from the mean of the
numbers `10, 9, 21, 16, 24?`
NDA Paper 1 2016

(A)

`5.2`

(B)

`5.0`

(C)

`4.5`

(D)

`4.0`

Solution:

Given, `x_(i) = 10, 9, 21, 16, 24`

`:. sum x_(i) = 10 + 9 + 21 + 16 + 24 = 80`

Now ` bar(X) = (sum x_(i))/n =(80)/5 = 16 => MD = (sum |x_(i) - bar(X)|)/n`

` (|10 - 16 | + |9-16| + | 21 -16| + | 16 -16| + | 24 -16|)/5`

`= (6+7+5+0+8)/5 = (26)/5 = 5.2`

Correct Answer is `=>` (A) `5.2`

Q 2127280181

If the total number of observations is `20, sum x_(i) = 1000` and
`sum x_(i)^(2) = 84000`, then what is the variance of the
distribution?
NDA Paper 1 2016

(A)

`1500`

(B)

`1600`

(C)

`1700`

(D)

`1800`

Solution:

Given, `N = 20, sum x_(i) = 1000` and `sum x_(i)^(2) = 84000`

Now , variance `= (sum x_(i)^(2) )/N - ((sum x_(i))/N )^(2) `

`= (84000)/(20) - (( 1000 )/(20))^(2)`

`= 4200- (50)^(2)`

`= 4200 - 2500 = 1700`

Correct Answer is `=>` (C) `1700`

Q 2117780680

The mean of the series `x_(1) , x_( 2) , ... , x_(n),`. is `bar(X)`. If `x_( 2)` is

replaced by `lamda` , then what is the new mean?
NDA Paper 1 2016

(A)

` bar (X) - x_(2) + lamda`

(B)

` (bar (X) - x_(2) - lamda)/n `

(C)

` (bar (X) - x_(2) + lamda)/n `

(D)

` (nbar (X) - x_(2) + lamda)/n `

Solution:

We know,` bar(X) = (x_(1) + x_(2) + ... + x_(n))/n`

` => x_(1) + x_(2) + ... + x_(n) = n bar(X)`

` => x_(1) + x_(3) + ... + x_(n) = n bar(X) - x_(2)`

` => x_(1) + x_(3) + ... + x_(n) + lamda`

` = n bar(X) - x_(2) + lamda`

`=> text( Mean) = text ( sum of all values)/ text(Total number of values)`

`= ( x_(1) + x_(3) + ... + x_(n) + lamda)/n`

` = (n bar(X) - x_(2) + lamda)/n`

Correct Answer is `=>` (D) ` (nbar (X) - x_(2) + lamda)/n `

Q 2167780685

For the data
`3, 5, 1, 6, 5, 9, 5, 2, 8, 6`
the mean, median and mode are `x, y` and `z`, respectively.
Which one of the following is correct ?
NDA Paper 1 2016

(A)

`x = y != z`

(B)

`x != y = z`

(C)

`x != y != z`

(D)

`x = y = z`

Solution:

Mean= ` (sum x_(i))/n`

`= (3+5+1+6+5+9+5+2 + 8+6)/(10)`

` =(50)/(10) = 5`

Now, the data in 'Ascending' order is;

`1,2,3,5,5,5,6,6,8,9`

Clearly, median (mid value) is `5` and mode (most

appeared value) is also `5`.

`x = y = z`

Correct Answer is `=>` (D) `x = y = z`

Q 2117880780

Consider the following statements in respect of a
histogram ,

1. The total area of the rectangles in a histogram is equal
to the total area bounded by the corresponding
frequency polygon and the X -axis.

2. When class intervals are unequal in a frequency
distribution the area of the rectangle is proportional
to the frequency.

which of the above statements is/ are correct?
NDA Paper 1 2016

(A)

Only `1`

(B)

Only `2`

(C)

Both `1` and `2`

(D)

Neither `1` nor `2`

Correct Answer is `=>` (C) Both `1` and `2`

Q 1619545410

The correlation coefficient between two variables
`X` and `Y` is found to be `0.6`. All the observations on
`X` and `Y` are transformed using the
transformations `U = 2 - 3 X` and `V = 4 Y + 1`. The
correlation coefficient between the transformed
variables `U` and `V` will be
NDA Paper 1 2015

(A)

`- 0.5`

(B)

`+ 0.5`

(C)

`- 0.6`

(D)

`+ 0.6`

Solution:

We have, `r(x, y) = 0.6`

To find `r(U, V),` where `U = 2 - 3x` and `V = 4y + 1`

Clearly ,` r(U, V) = ( cov(U,V))/sqrt(var(U) . var(V) )` ..........(1)

Now, `var(U) = var (2 - 3x) = (-3)^2 var(x) = 9 var(x)`

`var(V) = var ( 4 y + 1) = var( 4 y) = 16 var(y)`

`cov (U, V) = E [(U - bar(U)) (V- bar(V))]`

`= E [((2- 3x)- (2- 3barx)) ((4y + 1)- (4 bary + 14))]`

`= E [(-3 x + 3 barx) (4y - 4 bary)]`

`= (-3) (4) E [(x- bar x) (y- bar y)] = -12 cov(X, Y)`

Thus, from Eq. (i), we have

` r(U,V) = (-12cov(X,Y)) / sqrt( 9 var(x) . 16 var(y)) = (-12cov(x,y))/(3 . 4.sqrt(var(x) . var(y))`

` = - r (X, Y) =- 0.6`

Correct Answer is `=>` (C) `- 0.6`

Q 1649045813

The mean of five numbers is `30`. If one number is
excluded, their mean becomes `28`. The excluded
number is
NDA Paper 1 2015

(A)

`28`

(B)

`30`

(C)

`35`

(D)

`38`

Solution:

Let the numbers are `x_1. x_2. x_3. x_4` and `x_5` Then,

we have,

`(x_1 + x_2 + x_3 + x_4 + x_5)/5 = 30`

` x_1 + x_2 + x_3 + x_4 + x_5 = 150` .........(1)

Now, suppose `x_1` is excluded, then

` (x_2 + x_3 + x_4 + x_5)/4 = 28`(given)

`=> x_2 + x_3 + x_4 + x_5 = 112` .............(2)

From Eqs. (i) and (ii), we get

`:. x_1 = 150 - 112 = 38`

Correct Answer is `=>` (D) `38`

Q 1772480336

For two variables `x` and `y`, the two regression,
coefficients are `b_(yx) =- 3//2` and `b_(xy) =- 1//6`. The

correlation coefficient between `x` and `y` is
NDA Paper 1 2014

(A)

`-1//4`

(B)

`1//4`

(C)

`-1//2`

(D)

`1//2`

Solution:

Given that,

Two regression coefficients are,

`b_(xy) = - 3 // 2` and `b_(xy) = - 1 // 6`

Now, correlation coefficient between `x` and `y` i.e.,

` r =sqrt(b_(xy) . b_(yx) )`

` = sqrt( (-1// 6) xx (- 3//2) )`

` = sqrt(1/2 xx 1/2) = pm 1/2`

Here, we have to take negative sign because `b_(xy)` and `b_(yx)`

both have negative sign.

Hence, correlation coefficient `(r) =- 1/2`

Correct Answer is `=>` (C) `-1//2`

Q 1742580433

The variance of numbers `x_1, x_2, x_3 ,........., x_n` is `V`. Consider
the following statements.

I. If every `x_i` is increased by `2`, the variance of the new
set of numbers is `V`.
II. If the numbers `x_i` is squared, the variance of the new
set is `V^2`

Which of the following statements is/are correct?
NDA Paper 1 2014

(A)

Only 1

(B)

Only 2

(C)

Both 1 and 2

(D)

Neither 1 nor 2

Solution:

1. We know that, variance is not dependent on

change of origin. i.e., independent on change of origin.

So, if every `x_i` is increased b `y_2`, the variance of the new set of

numbers is not change i.e., `V`.

2. We know that, variance is dependent on change of scale.

So, if the number `x_i` is squared, the variance of the new set `v^2`.

i.e. If ` V(x_i) = V`

Then, `V_((x_i xx x_i)) = V _((x_i)) V_((x_i ))`

` = V.V`

` = V^2`

Correct Answer is `=>` (C) Both 1 and 2

Q 1742680533

What is the mean of the squares of the first `20` natural
numbers?
NDA Paper 1 2014

(A)

`151.5`

(B)

`143.5`

(C)

`65`

(D)

`72`

Solution:

Mean of first `20` natural numbers

` = text(Sum of the squares first of 20 natural numbers)/text(Number of observations)`

` = ((1^2 + 2^2 + 3^2 + ... + 20^2))/(20) quad [∵ sum n^2 = (n(n + 1) (2n + 1))/6 ]`

` = 1/(20) xx (20 (20 + 1) (20 xx 2 + 1))/6`

` = (21 xx (40 + 1))/6 = (21 xx 41 )/6 = (7 xx 41)/2 = (287)/2`

` = 143.5`

`:. ` Required mean `= 143.5`

Correct Answer is `=>` (B) `143.5`

Q 1732780632

`p, q, r, s` and `t` are five numbers such that the average of
`p, q` and `r` is `5` and that of `s` and `t` is `10`. What is the
average of all the five numbers?
NDA Paper 1 2014

(A)

`7.75`

(B)

`7.5`

(C)

`7`

(D)

`5`

Solution:

Given that, `p, q, r, s` and `t` are five numbers,

`:.` Average of `p, q` and `r = 5`

` => (p+q+r)/3 = 5`

`=> p + q + r =15` ......(1)

and average of `s` and `t = 10`

` => (s +t)/2 =10`

` => s + t = 20` ......(2)

Now, average of all five numbers

` = (p +q + r + s + t)/5`

` = ((p + q + r) + (s + t))/5`

` = (15+ 20)/5` [from Eq. (1) and (2)]

`= (35)/5 = 7`

Correct Answer is `=>` (C) `7`

Q 1730401312

If `barx` and `bary` are the means of two distributions such that
`barx < bary` and `bar z` is the mean of the combined distribution,
then which one of the following statements is correct?
NDA Paper 1 2014

(A)

`bar x < bar y < bar z`

(B)

`bar x > bar y > bar z`

(C)

` bar z = (bar x + bar y ) /2`

(D)

`bar x < bar z < bar y`

Solution:

It is obvious that , `bar x < bar z < bar y`

Correct Answer is `=>` (D) `bar x < bar z < bar y`

Q 1730501412

What is the mean deviation about the mean for the data
`4, 7, 8, 9, 10, 12, 13` and `17?`
NDA Paper 1 2014

(A)

`2.5`

(B)

`3`

(C)

`3.5`

(D)

`4`

Solution:

Mean deviation about the mean

` = (sum |x_i - bar x|)/N`

Here, `bar x = (4 + 7 + 8 + 9 + 10 + 12 + 13 + 17)/8 = 10`

`:.` Mean deviation about mean

` ( | 4 - 10 | + |7 - 10 | + |8-10 | + |9- 10 | + |10 -10| +| 12-10| + |13-10| +| 17-10 | )/8`

`= (6+3+2+1+0+2+3+7)/8 = (24)/8 = 3`

Correct Answer is `=>` (B) `3`

Q 1770601516

The variance of `20` observations is `5`. If each observation
is multiplied by `2`, then what is the new variance of the
resulting observations?
NDA Paper 1 2014

(A)

`5`

(B)

`10`

(C)

`20`

(D)

`40`

Solution:

New variance on scaling` = a^2 . old \ \ variance `

`a=2`

New variance `= 2^2 * 5 = 20`

Alternatively :

Let `x_1 , x_2 , ..... , x_(20)` be the given observations.

We have,

` 1/(20) sum_( i = 1)^(20) (x_i - bar x)^2 = 5`

To find Variance of `2x_1 , 2x_2 , 2x_3 , .... , 2x_(20)` . Let `bar x` denotes

the mean of new observation.

Clearly, `bar X = ( sum_( i = 1)^(20) 2x_i)/(20) = ( sum_( i = 1)^(20) x_i)/(20) = 2 barx`

Now, variance of new observation

` = 1/(20) sum_( i = 1)^(20) (2x_i - bar X)^2 = 1/(20) sum_( i = 1)^(20) (2x_i -2bar x)^2`

` = 1/(20) sum_( i = 1)^(20) 4(x_i - bar x)^2 = 4 ( 1/(20) sum_( i = 1)^(20) (x_i - bar x )^2 ) = 4 xx 5 = 20`

Correct Answer is `=>` (C) `20`

Q 1723656541

The number of telephone calls
received in `245` successive one minute intervals at an
exchange is given below in the following frequency
distribution.

What is the mean of the distribution?
NDA Paper 1 2014

(A)

`3.76`

(B)

`3.84`

(C)

`3.96`

(D)

`4.05`

Solution:

Given frequency distribution is

Mean `= (sum f_ix_i)/(sum f_i) = ([(0 xx 14 + 1 xx 21 + 2 xx 25 + 3 xx 43 + 4 xx 51+ 5 xx 40 + 6 xx 39+ 7 xx 12)])/((14+21+25+ 43+ 51+ 40+ 39+ 12)`

` = ( (0 + 21 + 50 + 129 + 204 + 200 + 234 + 84))/(245)`

` = (922)/(245) = 3.76`

Correct Answer is `=>` (A) `3.76`

Q 1733656542

The number of telephone calls
received in `245` successive one minute intervals at an
exchange is given below in the following frequency
distribution.

What is the median of the distribution?
NDA Paper 1 2014

(A)

`3.5`

(B)

`4`

(C)

`4.5`

(D)

`5`

Solution:

Here, `N/2 = (245)/2 = 122.5`

The cummulative frequency `154` which is qual or just

greater than `N/2 `

:. Required median= Value of the variable corresponding to

the cummulative frequency `154`

` = 4`

Correct Answer is `=>` (B) `4`

Q 1753656544

The number of telephone calls
received in `245` successive one minute intervals at an
exchange is given below in the following frequency
distribution.

What is the mode of the distribution?
NDA Paper 1 2014

(A)

`3`

(B)

`4`

(C)

`5`

(D)

`6`

Solution:

We see that in the frequency distribution the higher

frequency is `51`.

`:.` Required mode= Value of variable corresponding to the

higher frequency `= 4`

Correct Answer is `=>` (B) `4`

Q 1773656546

The mean and standard deviation
of `100` items are `50, 5` and that of `150` items are `40, 6`
respectively.

What is the combined mean of all `250` items?
NDA Paper 1 2014

(A)

`43`

(B)

`44`

(C)

`45`

(D)

`46`

Solution:

Given that, mean of `100` items i.e.,` bar x_(100) = 50`

Mean of `150` items i.e., ` bar x_(150) = 40`

and standard deviation of `100` items i.e., `sigma _(100) = 5`

Standard deviation of `150` items i.e., `sigma_(150) = 6`

Here, `n_1 = 100, bar x_(100) =50,`

and `n_2 = 150` and `bar x_(150) = 40`

:. Combined mean of all `250` items i.e.,

` bar x_(250) = ( n_1 . bar x_(100) + n_2 . bar x_(150) )/(n_1 + n_2)`

` = (100 xx 50+ 150 xx 40)/(100 + 150)`

` = (5000 + 6000)/(250) = (11000)/(250) = 44`

Correct Answer is `=>` (B) `44`

Q 1703656548

The mean and standard deviation
of `100` items are `50, 5` and that of `150` items are `40, 6`
respectively.

What is the combined standard deviation of all `250`
items?
NDA Paper 1 2014

(A)

`7.1`

(B)

`7.3`

(C)

`7.5`

(D)

`7.7`

Solution:

Given that, mean of `100` items i.e.,` bar x_(100) = 50`

Mean of `150` items i.e., ` bar x_(150) = 40`

and standard deviation of `100` items i.e., `sigma _(100) = 5`

Standard deviation of `150` items i.e., `sigma_(150) = 6`

We know that. If `n_1` and `n_2` are the sizes, `bar X_(100), bar X_(150)` are

the means and `sigma _(100) , sigma _(150)` are the standard deviation of the

series, then the standard deviation of the combined series is

` sigma = sqrt(( n_1 (sigma_(100)^2 + d_1^2 ) + n_2 (sigma_(150)^2 + d_2^2 ) )/(n_1 + n_2) )` ....(1)

where, `d_1 = barX_(100) - barX_(250)` and `d_ 2 = barX_(150) - barX_(250)`

Here, `d_ 1 =50 - 44 = 6 => d_1^2 =36`

and `d_ 2 = 40 - 44 = - 4 => d_2^2 = 16`

From Eq. (1),

Combined standard deviation of all `250` items i.e.,

` sigma_(250) = sqrt((100 {(5)^2 + 36} + 150 {(6)^2 + 16})/(100+ 150))`

` = sqrt((100 (25 + 36) + 150 (36 + 16))/(250))`

` = sqrt(100 xx 61 + 150 xx 52)/(250)`

` = sqrt(100 xx 61 + 15 xx 52)/5`

` = sqrt (610 + 780)/5 = sqrt (1390)/5`

` = (37.28)/5 = 7.456 = 7.5`

Correct Answer is `=>` (C) `7.5`

Q 1723756641

The mean and standard deviation
of `100` items are `50, 5` and that of `150` items are `40, 6`
respectively.

What is the variance of all the `250` items?
NDA Paper 1 2014

(A)

`50.6`

(B)

`53.3`

(C)

`55.6`

(D)

`59.3`

Correct Answer is `=>` (C) `55.6`

Q 1723178041

Let `X` denote the number of scores which exceed `4` in
`18` tosses of a symmetrical die. Consider the following
statements

I. The arithmetic mean of `X` is `6`.
II. The standard deviation of `X` is `2`.

Which of the above statements is/are correct?
NDA Paper 1 2014

(A)

Only 1

(B)

Only 2

(C)

Both 1 and 2

(D)

Neither 1 nor 2

Solution:

Given that, `n =` Total number of tosses `= 18`

and `X=` Number of Scores which exceed `4` in `18` tosses of

a symmetrical die `= { 5,6}`.

` => n(X) = 2`

`:. p = (n (X))/(n(s)) = 2/6 = 1/3`

and `q = 1 - p = 1 - 1/3 = 2/3 quad (∵ p + q = 1 )`

Now, arithmetic mean of `X = np`

` = 18 xx 1/3 = 6`

and standard deviation of `X= sqrt text(Variance of X)`

` = sqrt(npq)`

` = sqrt(18 xx 1/3 xx 2/3 )`

` = sqrt(4) = 2`

Hence, both statements are correct.

Correct Answer is `=>` (C) Both 1 and 2

Q 2317123989

The mean of 20 observations is 15. On checking,
it was found that two observations were wrongly
copied as 3 and 6. If wrong observations are
replaced by correct values 8 and 4, then the
correct mean is
NDA Paper 1 2013

(A)

`15`

(B)

`15.15`

(C)

`15.35`

(D)

`16`

Solution:

Given that

Mean of 20 observations `= 15`

`therefore` . Sum of 20 observations `= 20 xx 15 = 300`
`therefore` Sum of actual (correct) observations
`= 300-(3+ 6)+ (8+ 4)`
`= 300 - 9 + 12 = 303`

`therefore ` Correct mean = `303/20 = 15.15`

Correct Answer is `=>` (B) `15.15`

Q 2347134083

Consider the following statements
I. Both the regression coefficients have same
sign.
II. If one of the regression coefficients is greater
than unity, the other must be less than unity.

Which of the above statements is/are correct'?
NDA Paper 1 2013

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

Let `b_(y x ) > 1 => 1/b_(xy) < 1` ..............(i)

We have `b_(yx) * b_(xy) = r^2 le 1`

` = b_(xy) le 1/b_(yx) < 1`

[from Eq. (i)]

Hence, if one of the regression coefficients is greater than one.
the other must be less than one. So, both statements are true.

Correct Answer is `=>` (C) Both I and II

Q 2367134085

Which one of the following measures is
determined only after the construction of
commulative frequency distribution?
NDA Paper 1 2013

(A)

Arithmetic mean

(B)

Mode

(C)

Median

(D)

Geometric mean

Solution:

Median is determined only after the construction of
cummulative frequency distribution

Correct Answer is `=>` (C) Median

Q 2377134086

Coefficient of correlation is the measure of
NDA Paper 1 2013

(A)

central tendency

(B)

dispersion

(C)

Both central tendency and dispersion

(D)

Neither central tendency nor dispersion

Solution:

Coefficient of correlation is the measure of the intensity
or the magnitude of linear relationship between two variables

Correct Answer is `=>` (D) Neither central tendency nor dispersion

Q 2387134087

What is the variance of the first 11 natural
numbers?
NDA Paper 1 2013

(A)

(B)

(C)

(D)

Solution:

We know that. the variance of the first n natural numbers

`(n^2-1)/(12)`

`therefore ` Variance of the first 11 natural numbers

`((11)^2-1)/(12) = (121-1)/(12)`

`(120)/(12) = 10`

Correct Answer is `=>` (A) 10

Q 2317234180

Consider the following statements
I. Pie diagrams are suitable for categorical data.
II. The arc length of a sector of a pie diagram is
proportional to the value of the component
represented by the sector.

Which of the statements given above is/are correct?
NDA Paper 1 2013

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

In a pie chart, the arc length of each sector (and
consequently its central angle and area), is proportional to the
quantity it represents. While it is named for its resemblance to a
pie which has been sliced, there are variations on the way it can
be presented.
So, the both statements are correct.

Correct Answer is `=>` (C) Both I and II

Q 2357234184

Consider the following statements
I. The algebraic sum of the deviations of a set of
n values from its arithmetic mean is zero.
II. In the case of frequency distribution, mode is
the value of the variable which corresponds to
maximum frequency.
Which of the statements given above is/are correct?
NDA Paper 1 2013

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

I. Mathematically `Sigma (X-barX) = 0`

or for a frequency distribution

`Sigmaf(X-barX) = 0`

The algebraic sum of the deriations of the given set of
observations from their arithmetic mean is zero.
II. By definition of mode,
It is the value of the variable which corresponds to maximum
frequency.
So, both statements are correct

Correct Answer is `=>` (C) Both I and II

Q 2377234186

The variance of 20 observations is 5. If each
observation is multiplied by 2, then what is the
new variance of the resulting observations?
NDA Paper 1 2013

(A)

`5`

(B)

`10`

(C)

`20`

(D)

`40`

Solution:

Given that, Var `(X) = 5` ........(i)

We know that, Var `(Ax + b)= a^2Var(X)`
`= (2)^2 xx 5` [from eq (i)]
`= 4xx 5`
`=20`

(since, each observation is multiplied by 2)

Correct Answer is `=>` (C) `20`

Q 2317234189

The marks obtained bv 13 students in a test are 10,
3, 10, 12, 9, 7, 9, 6,7, 10, 8, 6 and 7, then the
median of this data is
NDA Paper 1 2013

(A)

`7`

(B)

`8`

(C)

`9`

(D)

`10`

Solution:

First we arrange the given data in ascending order,
we get
`3,6,6, 7, 7, 7,8,9,9, 10, 10, 10,12`
Total terms, `n = 13` (odd)

`therefore` Median = `((n+1)/2)th` term

` = ((13+1)/2)th` term = 7th term

` = 8`

Correct Answer is `=>` (B) `8`

Q 2307334288

Consider the following statements
I. Both variance and standard deviation are
measures of variability in the population.
II. Standard deviation is the square of the
variance.
Which of the above statements is/are correct?
NDA Paper 1 2013

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

Both variance and standard deviation are not the
measures of variability in the population.
Also, relation between variance and standard deviations

`sqrt(text(Variance)` = Standard deviation

`=> sqrtsigma = SD`

Hence, both statements are incorrect.

Correct Answer is `=>` (D) Neither I nor II

Q 2327434381

If `X` follows a binomial distribution with
parameters `n = 100` and `p = 1 / 3`, then `P(X = r)` is
maximum when
NDA Paper 1 2013

(A)

`r=16`

(B)

`r = 32`

(C)

`r = 33`

(D)

`r = 34`

Solution:

`P(X = r) = (r+ 1)th text(term of) (q + p)th`

`=> m = (n+1)/(1+p/q) = (101)/(1+2) = 33 2/3`

`[m] = 33`

Since `34` will be greatest.

`therefore X = 33`

Correct Answer is `=>` (C) `r = 33`

Q 2347434383

The Binomial distribution has
NDA Paper 1 2013

(A)

only one parameter

(B)

two parameters

(C)

three parameters

(D)

four parameters

Solution:

The Binomial distribution has two parameters `n` and `p`.
It is written as `X - B(n, p)` or `X ~ Bi (n, p)`.

Correct Answer is `=>` (B) two parameters

Q 2357534484

Consider the following frequency distribution

If the total of the frequencies is 100 and mode is
25, then which one of the following is correct?

NDA Paper 1 2013

(A)

`x=2y`

(B)

`2x=y`

(C)

`x = y`

(D)

`x = 3y`

Solution:

Given that, sum of frequencies `=100`
`=> 14+x+27+ y+15=100`
`=> x+ y+ 57= 100`
`=> x+ y=43` .............(i)

For mode `f_m =27, f_1 = x` and `f_2 = yf_1 = 20, h=10`

Clearly, `20-30` is the modal class,

Since, mode lies between `20-30`

`therefore` Mode = `f_1+ (f_m-f_1)/(2f_m-f_1-f_2) xx h`

given `25 = 20+(27-x)/(54-x-y)xx10`

`=> 5 = (270-10x)/(54 -x-y)`

`=> 270-10x = -5x-5y +270`

`=> 5x-5y = 0`

`therefore x =y` .............(ii)

From Eqs. (i) and (ii),

`2x = 43`

`therefore x = 43/2 = 21.5`
` = y = 21.5`

Correct Answer is `=>` (C) `x = y`

Set - 2

Q 2327734681

The average marks obtained by the students in a
class are 43. If the average marks obtained by
25 boys are 40 and the average marks obtained by
the girl students are 48, then what is the number
of girl students in the class?
NDA Paper 1 2013

(A)

`15`

(B)

`17`

(C)

`18`

(D)

`20`

Solution:

Given that, average marks obtained by the students in a
class `(barx_(BG)) = 43`
Total number of boys `(n_B ) = 25`
Average marks obtained by boys `(barx_B )= 40`
Average marks obtained t'y girls `(barx_G )= 48`
Let the total number of girls `= n_G`
Now, by formula

`barx_(BG) = (n_B * bar x_B +n_G * bar x_G)/(n_B+n_G)`

`=> 43 = (25 * 40+ n_G * 48)/(25+n_G)`

`1075+43 * n_G = 1000+48 * n_G`

`=> 75 = 5 n_G`

`n_G = 15`

`therefore` Required number of girls `= 15`

Correct Answer is `=>` (A) `15`

Q 2387734687

Marks obtained by 7 students in a subject are
30, 55, 75, 90, 50, 60, 39. The number of
students securing marks less than the mean marks
is
NDA Paper 1 2013

(A)

`7`

(B)

`6`

(C)

`5`

(D)

`4`

Solution:

Marks obtained by 7 students in a subject are `30, 55,
75, 90, 50, 60` and `39`.

`therefore` Mean marks = `(30+55+75+90+50+60+39)/7 = 399/7 = 57`

Hence, the number of students securing marks less than the
mean marks is 4 i.e.`,{30, 39, 50, 55}`.

Correct Answer is `=>` (D) `4`

Q 2357834784

Variance is always independent of the change of
NDA Paper 1 2013

(A)

origin but not scale

(B)

only scale

(C)

both origin and scale

(D)

None of these

Solution:

Variance is always independent of the change of origin
but not scale.
e.g .. Var `(ax+ b) = a^2V(x)`

Correct Answer is `=>` (A) origin but not scale

Q 2387034887

If two lines of regression are perpendicular, then
the correlation coefficient r is
NDA Paper 1 2013

(A)

`2`

(B)

`1/2`

(C)

`0`

(D)

None of these

Solution:

We know that, `tan theta = (1-r^2)/(|r|) *(sigma_x * sigma_y)/(sigma_x^2 + sigma_y^2)` Where `theta` is the angle between the two regression lines.
Given that, two lines of regression at perpendicular.
i.e.,`theta = 90^0`

`therefore tan90^0 = (1- r^2)/(|r|) * (sigma_x * sigma_y)/(sigma_x^2+sigma_y^2) = 1/0`

`=> |r| *( sigma_x^2+sigma_y^2) = 0`

`=> |r| = 0` (`because sigmax^2+sigmay^2 = 0)`

`therefore r = 0`

i.e., Correlation coefficient (r)= 0

Correct Answer is `=>` (C) `0`

Q 2337134982

The standard deviation of the observations `5, 5, 5,
5` and `5` is
NDA Paper 1 2013

(A)

`0`

(B)

`5`

(C)

`20`

(D)

`25`

Solution:

Given that, observations are `5, 5, 5, 5` and `5`.

Mean `(barx) = (5+5+5+5+5)/5 = 25/5 = 5`

Now `SD = sqrt(underset(i = 1) overset(5)Sigma (x_1- barx)^2/N)`

` = sqrt((5-5)^2+(5-5)^2+(5-5)^2+(5-5)^2+(5-5)^2)/5`

` = sqrt((0+0+0+0+0)/5)`

` = 0`

Correct Answer is `=>` (A) `0`

Q 2377145086

What is the mean of first `n` odd natural numbers?
NDA Paper 1 2012

(A)

`n`

(B)

`(n + 1)//2`

(C)

`n(n + 1)//2`

(D)

`n + 1`

Solution:

` = 1 + 3 + 5 + 7 + 9 + ··· + n` term

` = n/2 [2 xx 1 + (n - 1)2 ] = n/2 xx 2n = n^2`

`:.` Mean ` = text (Sum of an odd natural numbers)/text( total numbers) = n^2/n = n`

Correct Answer is `=>` (A) `n`

Q 2317145089

If the arithmetic mean of numbers `a, b, c, d` and `e` is
`M`, then what is the value of `(a - M) + (b - M)`
`+(c - M) + (d - M) + (e - M)`?
NDA Paper 1 2012

(A)

`M`

(B)

`a + b + c + d + e`

(C)

`0`

(D)

` 5 M`

Solution:

Given. `( a+b+c+d+e)/5 = M`

`=> a + b + c + d + e = 5M`

`:. (a - M) + (b - M) + (c - M) + (d - M) + (e - M)`

`= (a + b + c + d + e) - 5M`.

` = 5M - 5M = 0`

Correct Answer is `=>` (C) `0`

Q 2377134986

The mean of 10 observations is 5. If 2 is added to
each observation and then multiplied by 3, then
what will be the new mean?
NDA Paper 1 2012

(A)

`5`

(B)

`7`

(C)

`15`

(D)

`21`

Solution:

Let observations are `x_1,x_2 . ... x_(10)`

Given `(x_1+x_2+x_3+ ...............x_(10))/(10) = 5`

`=> x_1+x_2+x_3+ ...................x_(10) = 50`

Again, according to the question
New mean

` = ([(x_1+2)+(x_2+2)+(x_3+2)+ ..................+(x_(10)+2)] xx3)/(10)`

` = [(x_1+x_2+x_3 ...............x_(10)+20] xx3)/(10)`

` = ((50+20)xx3)/(10) = (70xx3)/(10) = 21` [from Eq (i)]

Correct Answer is `=>` (D) `21`

Q 2317145080

The algebraic sum of the deviations of `20`
observations measured from `30` is `2`. What would
be the mean of the observations'?
NDA Paper 1 2012

(A)

`30`

(B)

`32`

(C)

`30.2`

(D)

`30.1`

Solution:

According to the question

`= underset(i = 1) overset(20)Sigma(x_i-30) = 2` (given)

`=> underset(i = 1) overset(20) Sigma x_i-600 = 2 => underset(i = 1) overset(20) Sigmax_i = 602`

`therefore` Mean = `(underset(i =1) overset(20) Sigma x_i)/(20) = (602)/(20) = 30.1`

Correct Answer is `=>` (D) `30.1`

Q 2347145083

The median of `2 7` observations of a variable is `18`.
If three more observations are made and the
values of these observations are `16, 18` and `50`,
then what is the median of these `30` observations?
NDA Paper 1 2012

(A)

`18`

(B)

`19`

(C)

`25.5`

(D)

Cannot be determined

Solution:

Median of `27` observations `= 18`
Then `((27+1)/2)th` observation `=18`

`14th` observation `= 18`

Then, observation whose value is 16 come before 18, i.e ..
14th observation.
Now, number of observation is even 30 Observations

Median = `((30/2)th+(30/2+1)th)/2`

` = (15 th text(observation + 16th observation))/2`

` = (18+18)/2 = 18`

Correct Answer is `=>` (A) `18`

Q 2387145087

Frequency curve may be
NDA Paper 1 2012

(A)

symmetrical

(B)

positive skew

(C)

negative skew

(D)

All of these

Solution:

Frequency curve may be symmetrical, positive skew
and negative skew.
For symmetry

`=> text(Mean = Median = Mode)`

`=> barx = M_d = M_o`

For positive skew, Mean > Median > Mode
`barx > M_d > M_o`

For negative skew,
Mean < Median < Mode

Correct Answer is `=>` (D) All of these

Q 2317245180

If the mean of few observations is `40` and standard
deviation is `8`, then what is the coefficient of
variation?
NDA Paper 1 2012

(A)

`1%`

(B)

`10%`

(C)

`20%`

(D)

`30%`

Solution:

The coefficient of variation

` = 100xx sigma/(barx) = 100 xx 8/40 = (800)/40 = 20`

Correct Answer is `=>` (C) `20%`

Q 2337245182

What is the standard deviation of `7, 9, 11, 13, 15`
NDA Paper 1 2012

(A)

`2.4`

(B)

`2.5`

(C)

`2.7`

(D)

`2.8`

Solution:

Mean `(barx) = (7+9+11+13+15)/5 = 55/5 = 11`

`Sigma d_i^2 = 40`

`therefore` Variance `(sigma^2) = (Sigmad_i^2)/n = 40/5 = 8`

Standard deviation = `sqrt8 = 2.8`

Correct Answer is `=>` (D) `2.8`

Q 2347245183

Which one of the following is a measure of
dispersion?
NDA Paper 1 2012

(A)

Mean

(B)

Median

(C)

Mode

(D)

Standard deviation

Solution:

Standard deviation is a measure of dispersion.

Correct Answer is `=>` (D) Standard deviation

Q 2367245185

What is the mode for the data `20, 20, 20, 21, 21,
21,21,21,22,22,22,22,22,22,22,23,23,23,
23, 23, 24, 24` and `25?`
NDA Paper 1 2012

(A)

`7`

(B)

`21`

(C)

`22`

(D)

`25`

Solution:

The given observations are

Mode = Higher frequency of observation = 22 (7 times)

Correct Answer is `=>` (C) `22`

Q 2307245188

Consider the following statements
I A continuous random variable can take all
values in an interval.
II A random variable which takes a finite number
of values is necessarily discrete.
III Construction of a frequency distribution is
based on data which are discrete.
Which of the above statements are correct?
NDA Paper 1 2012

(A)

I and II

(B)

II and Ill

(C)

I and Ill

(D)

All of these

Solution:

Here, Statement Ill is wrong because construction of a
frequency distribution is based on data which are both discrete
as well as continuous.

Correct Answer is `=>` (A) I and II

Q 2327345281

Consider the following statements
I Two independent variables are always
uncorrelated.
II The coefficient of correlation between two
variables X and Y is positive when X decreases,
then Y decreases.
Which of the above statements is/are correct?
NDA Paper 1 2012

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor Ill

Solution:

I. If two variables are independent, then there is
uncorrelation i.e., r = 0.
II. The coefficient of correlation between two variables X and Y is
positive. When X positive, then Y positive

Correct Answer is `=>` (A) Only I

Q 2357345284

If a variate `X` takes values `2 ,9, 3, 7, 5, 4, 3, 2` ·and
`10`, then what is the median ?
NDA Paper 1 2012

(A)

`2`

(B)

`4`

(C)

`7`

(D)

`9`

Solution:

The given observations are arranged in ascending
order
`2,2,3,3, 4,5, 7,9, 10`
Here, total term = `9`

`therefore` Median = `((9+1)/2) th ` term = `(10/2) th` term

` = 5th` term = `4`

Correct Answer is `=>` (B) `4`

Q 2317445380

Some measures of central tendency for n discrete
observations are given below
I. Arithmetic mean II. Geometric mean
III. Harmonic mean IV Median
A desirable property of a measure of central
tendency is if every observation is multiplied by c,
then the measure of central tendency is also
multiplied by c, where c > 0. Which of the above
measures satisfy the property'?
NDA Paper 1 2011

(A)

I, II and Ill

(B)

I. II and IV

(C)

Ill and IV

(D)

I. II, Ill and IV

Solution:

Let a and b be two observations. then

I. Arithmetic mean, `AM = ((a+b)/2)`

Now, we multiply by c tn every observation, then

`AM = (ac+bc)/2 = c * ((a+b)/2)`

II. Geometric mean. `GM = sqrt(ab)`

Now, we taken ac and be, then

`GM = sqrt(ac * bc) = c * sqrt(ab)`

Ill. Harmonic mean. `HM = (2ab)/(a+b)`

Now, we take, ac and be, then

`HM = (2(ac)*(bc))/(ac+bc) = (2c^2ab)/(c(b+a))`

`HM = c * ((2ab)/(a+b))`

IV. Median, `M_e`
(i) Let a, b. d. t and h be five observations, then
total term `= 4 + 1 = 5 = n`

Median = `((n+1)/2) th` term = `((5+1)/2)th` term

= 3rd term = d

(ii) Let a, b, d and f be four observations, then total number of
terms= 4 = n

Median = `(n/2)th` term+` (n/2+1)th` term

= 2nd term+3rd term = (b+d)

Now, we take, ac, bc , dc and fc, then

Median = `(n/2)th` term +`(n/2+1)th` term

= 2nd term+ 3 rd term

` = bc+dc = c*(b+d)`

Correct Answer is `=>` (D) I. II, Ill and IV

Q 2347445383

If a variate `X` takes values `2, 3, 4, 2, 5, 4, 3, 2` and
`1`, then what is the mode?
NDA Paper 1 2011

(A)

`2`

(B)

`3`

(C)

`4`

(D)

`5`

Solution:

The values of variates `3, 4, 2, 5, 4, 3, 2` and `1`.
The higher frequency value, which takes variate is mode `= 2`

Correct Answer is `=>` (A) `2`

Q 2377445386

Which one of the following is the mean of the
data given below?
NDA Paper 1 2011

(A)

`17`

(B)

`18`

(C)

`19`

(D)

`20`

Solution:

`therefore` Mean = `(Sigmaf_ix_i)/(Sigmaf_i) = (12+40+98+216+192+112+90)/(2+4+7+12+8+4+3)`

` =(760)/(40) = 19`

Correct Answer is `=>` (C) `19`

Q 2317445389

Students of three sections of a class, having `30, 30`
and `40` students appeared for a test of `100` marks.
If the arithmetic means of the marks of the three
sections are `72.2, 69` and `64.1` in that order, then
what is the arithmetic mean of the marks of all the
students of the three sections?
NDA Paper 1 2011

(A)

`66.6`

(B)

`67.3`

(C)

`68`

(D)

`70.6`

Solution:

let A, B and C be the sections of a class having `30, 30`
and `40` students. respectively.
Also, given that the students of each section securing the
arithmetic means of the marks `72.2, 69` and `64.1`, respectively.
Now, the total marks secured by the students of section A
=` 30 xx 72.2 = 2166`
The total marks secured by the students of section B
`= 30 xx 69 = 2070`
and the total marks secured by the students of section C
=` 40 xx 64.1 = 2564`
So, the arithmetic mean of marks of all the students of three
sections

` =(2166+2070+2564)/(100) = (6800)/(100) = 68`

Correct Answer is `=>` (C) `68`

Q 2357545484

If the variance of the data `2, 4, 5, 6` and `17` is `v`,
then what is the variance of the data `4, 8, 10, 12`
and `34?`
NDA Paper 1 2011

(A)

`v`

(B)

`4v`

(C)

`v^2`

(D)

`2v`

Solution:

We know that

`Var (lamdax) = lamda^2 Var (x)` ..............(i)

Given data `x = 2, 4, 5, 6, 17` and its variance,
`Var(x) = v`
Now, multiply by 2 in above data numbers

`x = 4 , 8 , 10 ,12 , 34`

Its variance, `Var(2x) = (2)^2 Var(x) = 4v`

Correct Answer is `=>` (B) `4v`

Q 2337645582

Study the following table and
answer the questions that follow.
What is the total population for the year 1997?
NDA Paper 1 2011

(A)

`810`

(B)

`830`

(C)

`970`

(D)

`1030`

Solution:

The total population for the year 1997 = 810

Correct Answer is `=>` (A) `810`

Q 2337645582

Study the following table and
answer the questions that follow.
What is the female urban population in the year 1995?
NDA Paper 1 2011

(A)

`390`

(B)

`410`

(C)

`430`

(D)

`470`

Solution:

The female urban population in the year 1995 = 410

Correct Answer is `=>` (B) `410`

Q 2337645582

Study the following table and
answer the questions that follow.
What is the urban population in the year 1 997'?
NDA Paper 1 2011

(A)

`400`

(B)

`460`

(C)

`490`

(D)

`510`

Solution:

The total number of urban population in the year 1997
= 310 + 180= 490

Correct Answer is `=>` (C) `490`

Q 2337645582

Study the following table and
answer the questions that follow.
What is the total population in the year 1998?
NDA Paper 1 2011

(A)

`1000`

(B)

`1020`

(C)

`1040`

(D)

`1050`

Solution:

The total population in the year 1998 = 1050

Correct Answer is `=>` (D) `1050`

Q 2337645582

Study the following table and
answer the questions that follow.
What is the difference between the number of
females and the number of males in the year
1995?
NDA Paper 1 2011

(A)

`90`

(B)

`100`

(C)

`110`

(D)

`120`

Solution:

The difference between the number of females and the
number of males in the year 1995 = (630 -720) = 90

Correct Answer is `=>` (A) `90`

Q 2337645582

Study the following table and
answer the questions that follow.
In which year is the male population minimum?
NDA Paper 1 2011

(A)

`1995`

(B)

`1996`

(C)

`1997`

(D)

`1998`

Solution:

In year 1997, the male population is minimum which is 440.

Correct Answer is `=>` (C) `1997`

Q 2337645582

Study the following table and
answer the questions that follow.
In which year is the female population maximum?
NDA Paper 1 2011

(A)

`1995`

(B)

`1996`

(C)

`1997`

(D)

`1998`

Solution:

In year 1995, the female population is maximum which is
720.

Correct Answer is `=>` (A) `1995`

Q 2337645582

Study the following table and
answer the questions that follow.
What is the percentage of rural male population
(over the whole population) in the year I 998?
NDA Paper 1 2011

(A)

`80/3 %`

(B)

`100/3 %`

(C)

`35%`

(D)

`40%`

Solution:

The percentage of rural male population (over the whole
population) in the year 1998

` = (280)/(1060)xx100`

` = (2800)/(106) = (1400)/(53) = 26.4 %`

` = (80)/3 %` approx

Correct Answer is `=>` (A) `80/3 %`

Q 2367745685

Study the pie chart given below
and answer the questions that follow.

The following pie chart gives the distribution of funds in a
five yea.r plan under the major heads of development
expenditures
Agriculture (A), Industry (B), Education (C),
Employment (D) and Miscellaneous (E).
Which head is allocated the maximum funds'?
NDA Paper 1 2011

(A)

Agriculture

(B)

Industry

(C)

Employment

(D)

Miscellaneous

Solution:

Employment is allocated maximum funds.

Correct Answer is `=>` (C) Employment

Q 2367745685

Study the pie chart given below
and answer the questions that follow.

The following pie chart gives the distribution of funds in a
five yea.r plan under the major heads of development
expenditures
Agriculture (A), Industry (B), Education (C),
Employment (D) and Miscellaneous (E).
How much money (in crore) is allocated to Education'?
NDA Paper 1 2011

(A)

3000

(B)

6000

(C)

9000

(D)

10800

Solution:

Money allocated to Education ` = (30^0)/(360^0) xx36000 = 3000`

Correct Answer is `=>` (A) 3000

Q 2367745685

Study the pie chart given below
and answer the questions that follow.

The following pie chart gives the distribution of funds in a
five yea.r plan under the major heads of development
expenditures
Agriculture (A), Industry (B), Education (C),
Employment (D) and Miscellaneous (E).
How much money (in crore) is allocated to both
Agriculture and Employment?
NDA Paper 1 2011

(A)

20000

(B)

21000

(C)

24000

(D)

27000

Solution:

Money allocated to both Agriculture and Employment

` = (90^0+120^0)/(360^0)xx36000`

` = (210)/(360) xx 36000 = 21000`

Correct Answer is `=>` (B) 21000

Q 2367745685

Study the pie chart given below
and answer the questions that follow.

The following pie chart gives the distribution of funds in a
five yea.r plan under the major heads of development
expenditures
Agriculture (A), Industry (B), Education (C),
Employment (D) and Miscellaneous (E).
How much excess money (in crore) is allocated to
Miscellaneous over Education?
NDA Paper 1 2011

(A)

3600

(B)

4200

(C)

4500

(D)

4800

Solution:

Required value of money ` = (75-30)/(360^0) xx36000`

` = (45)/(360)xx36000 = 4500`

Correct Answer is `=>` (C) 4500

Q 2357845784

What is the median of the distribution `3, 7, 6, 9,
5, 4` and `2?`
NDA Paper 1 2011

(A)

`5`

(B)

`6`

(C)

`7`

(D)

`8`

Solution:

Firstly, we arrange the given observation in ascending
order
`2, 3, 4, 5, 6, 7` and `9`
Total terms, `n = 7`

So Median = `((n+1)/2)th` term =` ((7+1)/2) th` term

= 4th term = 5

Correct Answer is `=>` (A) `5`

Q 2307845788

The mean of 7 observations is 10 and that of 3 observations is 5. What is the mean of all the 10 observations?
NDA Paper 1 2011

(A)

`15`

(B)

`10`

(C)

`8.5`

(D)

`7.5`

Solution:

Given mean of 7 observations = 10

`=> (underset(i = 1)overset(7)Sigmax_i)/7 = 10 => underset(i = 1)overset(7)Sigmax_i = 70`..............(i)

and the mean of 3 observations = 5

` (underset(i = 1)overset(3)Sigmax_i)/3 = 5 => underset(i = 1)overset(3)Sigmax_i = 15`........(ii)

On adding Eqs. (i) and (ii), we get

`underset(i = 1)overset(7)Sigmax_i+ underset(i = 1)overset(3)Sigmax_i = 70+15 => underset(i = 1)overset(10)Sigmax_i = 85`

`therefore` Mean of 10 observations ` = (underset(i = 1)overset(10)Sigmax_i)/(10) = 85/10 = 8.5`

Correct Answer is `=>` (C) `8.5`

Q 2347045883

Consider the following statements in respect of
the above frequency distribution
I. The median is contained in the modal class.
II. The distribution is bell-shaped.
Which one of the above statements is/are correct?
NDA Paper 1 2011

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

`N=21`

`N/2 = 21/2 = 10.5`

Since. the median class is `10.5-15.5`

`therefore` Median = `10.5+(10.5-10)/6xx5`

` = 10.5+0.417 = 10.917`

Thus, median is not contained in the modal class and the
distribution is not bell-shaped because in this distribution
Mean `ne` Median `ne` Mode

Correct Answer is `=>` (D) Neither I nor II

Q 2377045886

The following table gives the
continuous frequency distribution of a continuous
variable X.
What is the median of the above frequency
distribution'?
NDA Paper 1 2011

(A)

(B)

(C)

(D)

Solution:

`therefore N/2 = 50/2 = 25`

Since. median group is`20-30`

`therefore` Median = `20+(25-15)/(20)xx10`

` = 20+5 = 25`

Correct Answer is `=>` (C) 25

Q 2377045886

The following table gives the
continuous frequency distribution of a continuous
variable X.
What is the mean of the above frequency distribution?
NDA Paper 1 2011

(A)

(B)

(C)

(D)

Solution:

`therefore N/2 = 50/2 = 25`

Mean = `(Sigmafx)/(Sigmaf) = 1300/50 = 26`

Correct Answer is `=>` (B) 26

Q 2337145982

Consider the following statements with regard to
correlation coefficient r between random variables
x and_y
I. `r = + 1` or `-1` means there is a linear relation
between x and y'·
II. `- 1 le r le 1` and `r^2` is a measure of the linear
relationship between the variables.

Which of the statements given above is/are correct?
NDA Paper 1 2011

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

Both the statements I and II are correct. by property of
correlation coefficient

Correct Answer is `=>` (C) Both I and II

Q 2357145984

If the values of a set are measured in cm, what will
be the unit of variance?
NDA Paper 1 2011

(A)

`cm`

(B)

`cm^2`

(C)

`cm^3`

(D)

No unit

Solution:

If tile values of a set are measured in em, then the unit
of variance is `cm^2`

Correct Answer is `=>` (B) `cm^2`

Q 2327156081

What is the geometric mean of the data `2, 4, 8, 16`
and `32?`
NDA Paper 1 2011

(A)

`2`

(B)

`4`

(C)

`8`

(D)

`16`

Solution:

Required geometric mean ` = root(5)(2 * 4 * 8 * 16 * 32)`

` = root(5)(2^(1+2+3+4+5))`

` = root(5)(2^(15))`

`2^3 = 8`

Correct Answer is `=>` (C) `8`

Q 2347156083

What is the cumulative frequency curve of
statistical data commonly called?
NDA Paper 1 2011

(A)

Cartogram

(B)

Histogram

(C)

Ogive

(D)

Pictogram

Solution:

The cumulative frequency curve of statistical data is
called ogive.

Correct Answer is `=>` (C) Ogive

Q 2357256184

The arithmetic mean of two numbers exceeds
their geometric mean by `2` and the geometric
mean exceeds their harmonic mean by `1.6`. What
are the two numbers?
NDA Paper 1 2010

(A)

16 and 4

(B)

81 and 9

(C)

256 and 16

(D)

625 and 25

Solution:

Let `H` be the harmonic mean of two numbers.

`:. A = G + 2, G = H + 1 .6`

and `A = H + 1.6 + 2 = H + 3.6`

We know that, `AH = G^2`

`(H + 3.6)H = (H + 1 .6)^2 => H^ 2 + 3.6H = H^2 + 2.56 + 3.2H`

`=> H = (2.56)/(0.4) = 6.4`

`:. A = 6.4 + 3.6 = 10` and `G = 6.4 + 1.6 = 8`

Let two numbers be a and b.

`:. a + b = 20` ... (i)

and `ab = 64` ... (ii)

We know that, `(a- b)^2 =(a+ b)^2 - 4ab = 400 - 256 = 144`

`=> a - b = 12` ... (iii)

On solving Eqs. (i) and (iii), we get

`a= 16` and `b = 4`

Correct Answer is `=>` (A) 16 and 4

Q 2327256181

What is the mean deviation of the data `2, 9, 9, 3,
6, 9` and `4?`
NDA Paper 1 2010

(A)

`2.23`

(B)

`2.57`

(C)

`3.23`

(D)

`3.57`

Solution:

`because` Mean ` (2+9+9+3+6+9+4)/7`

` = 42/7 = 6 = barx`

`therefore ` Mean deviation =` (Sigma|x- barx|)/n`

` = (|2-6|+3|9-6|+|3-6|+|6-6|+|4-6|)/7`

` = (4+9+3+0+2)/7 = 18/7 = 2.57`

Correct Answer is `=>` (B) `2.57`

Q 2307256188

If a set of `n` values `x_1, x_2 , ... , X_n` has standard
deviation `sigma`, then what is the standard deviation of
`n` values `x_1 + k, x_2 + k, ... , x_n + k?`
NDA Paper 1 2010

(A)

`sigma`

(B)

`sigma+k`

(C)

`sigma-k`

(D)

`ksigma`

Solution:

We know that, if a number is added in values, then the
standard deviation remains unaltered.
`therefore` `SD` of new values `= sigma`

Correct Answer is `=>` (A) `sigma`

Q 2307356288

The two lines of regression are `8x - 1 0y = 66` and
`40x - 18 y = 214` and variance of `x` series is `9`.
What is the standard deviation of `y` series ?
NDA Paper 1 2010

(A)

`3`

(B)

`4`

(C)

`6`

(D)

`9`

Solution:

For the line `8x - 10y = 66, r_1 = 4/5`

and for the line `40x -18y = 214`

`r_2 = 9/20` and `sigma_x^2 = 9`

`sigma_x = 3 => r^2 = 36/100` (`because r = sqrt(r_1 *r_2)`)

`=> r = 0.6`

`therefore sigma_y = (r_1xx sigma_x)/r = (4/5xx3)/0.6 = 4/5xx3xx10/6 = 4`

Correct Answer is `=>` (B) `4`

Set - 3

Q 2327556481

The standard deviation of some consecutive
integers is found to be 2. Which of the following
statements best describes the nature of the
consecutive integers?
NDA Paper 1 2010

(A)

The integers are any set of eight consecutive integers

(B)

The integers are any set of eight consecutive positive integers

(C)

The integers are any set of seven consecutive integers

(D)

None of the above

Solution:

Since, the standard deviation of same consecutive
inte!Jers is 2, then these integers are any set of seven
consecutive integers

Correct Answer is `=>` (C) The integers are any set of seven consecutive integers

Q 2367556485

Consider the following data

What is the variability in the wages of the workers
in Factory A?
NDA Paper 1 2010

(A)

100% more than the variability in the wages of the workers in factory B

(B)

50% more than the variability in the wages of the workers in factory B

(C)

50% less than the variability in the wages of the workers in factory B

(D)

150% more than the variability in the wages of the workers in factory B

Solution:

The variability in the wages of the workers in factory A is
50% more than the variability in the wages of the workers in
factory B.

`because` Coefficient of variation = `(SD)/text(Mean) xx100`

For factory A `(40.50)/(540)xx100 = 7.5`

and for factory B `(31)/(620)xx100 = 5`

Correct Answer is `=>` (B) 50% more than the variability in the wages of the workers in factory B

Q 2307556488

If the distributions `x` and `y` with total number of
observations `36, 64` and mean `4, 3` respectively are
combined, then what is the mean of the resulting
distribution `x + y ?`
NDA Paper 1 2010

(A)

`3.26`

(B)

`3.32`

(C)

`3.36`

(D)

`3.42`

Solution:

Required mean = `(36xx4+64xx3)/(36+64) = (144+192)/(100)`

` = (336)/(100) = 3.36` (`because barx_(12) = ((barx_1n_1+barx_2n_2)/(n_1+n_2))`

Correct Answer is `=>` (C) `3.36`

Q 2347656583

Consider the following data

What is the regression equation of y on `x` ?
NDA Paper 1 2010

(A)

`y = 0.6 + 0.4x`

(B)

`y = 0.7 + 0.3x`

(C)

`y = 6 + 5x`

(D)

`y = 4 + 9x`

Solution:

`barx = 30/5 = 6 ` and `bary = 15/5 = 3`

`therefore b_(yx) = (nSigmaxy -(Sigmax)(Sigmay))/(nSigmax^2- (Sigmax)^2)`

` = (5xx94-30xx15)/(5xx190-(30)^2)`

` = (470-450)/(950-900)`

` = 20/50 = 2/5 = 0.4`

`y -3 = 0.4(x-6)`

` => y = 0.4x+3-2.4`

`y = 0.4x+0.6`

Correct Answer is `=>` (A) `y = 0.6 + 0.4x`

Q 2307656588

The frequency distribution of life
of 90 TV tubes whose median life is 1 7 months is as follow
What is the lower limit of the median class?
NDA Paper 1 2010

(A)

`10`

(B)

`15`

(C)

`20`

(D)

`25`

Solution:

`because N = 90`

`therefore N/2 = 45`

Here, the lower limit of median class is 15.

Correct Answer is `=>` (B) `15`

Q 2307656588

The frequency distribution of life
of 90 TV tubes whose median life is 1 7 months is as follow
What is the missing frequency y?
NDA Paper 1 2010

(A)

(B)

(C)

(D)

Solution:

`because N = 90`

`therefore N/2 = 45`

Given that median = 17

`because M = l+{N/2-C}/f xxh`

` => 17 = 15+{45-(15+x)}/(35) xx 5`

`=> 17 = 15+(30-x)/7`

` => 119 = 105+30-x`

`=> x = 16`

`therefore x+y = 36`

`=> y = 36-16 = 20`

`therefore` Unknown frequency = 20

Correct Answer is `=>` (A) 20

Q 2307656588

The frequency distribution of life
of 90 TV tubes whose median life is 1 7 months is as follow
What is the cumulative frequency of the modal
class?
NDA Paper 1 2010

(A)

(B)

(C)

(D)

Cannot be determined

Solution:

`because N = 90`

`therefore N/2 = 45`

.Cumulative frequency of modal class
`=50+ x=' 50+ 16 = 66`

Correct Answer is `=>` (C) 66

Q 2327856781

The table below gives an
incomplete frequency distribution with two missing
frequencies `f_1` and `f_2` .

The total frequency is 18 and the arithmetic mean
of x is 2.
What is the value of `f_2 ?`
NDA Paper 1 2010

(A)

`4`

(B)

`3`

(C)

`2`

(D)

`1`

Solution:

Now `f_1+f_2+11 = 18`

`=> f_1+f_2 = 7`

On putting the value of `f_2` in Eq. (i), we get

`f_1 = 7-4 = 3`

Arithmetic mean

`barx = (Sigmafx)/(Sigmaf)`

` 2 = (0+f_2+8+12+12)/(18) => 36 = f_2+32 => f_2 = 4`

Correct Answer is `=>` (A) `4`

Q 2327856781

The table below gives an
incomplete frequency distribution with two missing
frequencies `f_1` and `f_2` .

The total frequency is 18 and the arithmetic mean
of x is 2.
What is the standard deviation?
NDA Paper 1 2010

(A)

`sqrt5/2`

(B)

`sqrt5/3`

(C)

`4/3`

(D)

`16/9`

Solution:

Now `f_1+f_2+11 = 18`

`=> f_1+f_2 = 7`

On putting the value of `f_2` in Eq. (i), we get

`f_1 = 7-4 = 3`

`because barx = 2`

` = (0+4+8+12)/(18)` and `f_1+f_2 = 7`

`=> f_1 = 3`

Now `SD = sqrt((Sigmaf(x-barx)^2)/N)`

`sigma = sqrt((32)/(18) = sqrt(16/9) = 4/3`

Correct Answer is `=>` (C) `4/3`

Q 2327856781

The table below gives an
incomplete frequency distribution with two missing
frequencies `f_1` and `f_2` .

The total frequency is 18 and the arithmetic mean
of x is 2.
What is the coefficient of variance?
NDA Paper 1 2010

(A)

`200/3`

(B)

`(50 sqrt5)/9`

(C)

`600/sqrt5`

(D)

`150`

Solution:

Now `f_1+f_2+11 = 18`

`=> f_1+f_2 = 7`

On putting the value of `f_2` in Eq. (i), we get

`f_1 = 7-4 = 3`

Coefficient of variance = `sigma/barx xx 100`

` = 4/3xx1/2xx100 = 200/3`

Correct Answer is `=>` (A) `200/3`

Q 2307056888

A class consists of 3 sections A, B and C with 35,
35 and 30 students respectively. The arithmetic
means of the marks secured by students of
sections A and B, who appeared for a test of 100
marks are 7 4 and 70 respectively. The arithmetic
mean of the marks secured by students of section
C, who appeared for a test in the same subject
which carried 7 5 marks is 51. What is the average
percentage of marks secured by all the 100
students of the three sections?
NDA Paper 1 2009

(A)

`70`

(B)

`70.8`

(C)

`65`

(D)

`67.5`

Solution:

Since, section `C` carried `51` average marks of `75`.
`therefore` Section `C` carried = `51/75xx100 = 68` average marks out of 100

`therefore` Average percentage marks = `(35xx74+35xx70+30xx68)/(100)`

` = (2590+2450+2040)/(1000`

` = 70.80`

Correct Answer is `=>` (B) `70.8`

Q 2387467387

In a study on the relationship between investment
(x) and profit (y), the following two regression
equations were obtained based on the data on x
and y.

`3x + y -12 = 0`
`x+2y-14=0`
What is the mean `barx`' ?
NDA Paper 1 2009

(A)

`6`

(B)

`5`

(C)

`4`

(D)

`2`

Solution:

Since, lines of regression passes through `(barx, bary)`

`therefore 3barx+bary-12 = 0` .........(i)

and `barx+2 bary-14 = 0` .................(ii)

on solving Eqs (i) and(ii) we get

`barx = 2 ` and `bar y = 6`

Correct Answer is `=>` (D) `2`

Q 2317567489

Following table gives the mean and variance of
monthly demand for four products A, B, C and D
in a supermarket

For which product the demand is consistent?
NDA Paper 1 2009

(A)

Product A

(B)

Product B

(C)

Product C

(D)

Product D

Solution:

Since, coefficient of varince = `(SD)/text(Mean)`

Coefficient of variance `A = sqrt(12)/(60) = 3.46/(60) = 0.057`

Coefficient of variance B = `sqrt(25)/90 = 5/90 = 0.056`

Coefficient of variance C = `sqrt(36)/80 = 6/80 = 0.075`

Coefficient of variance D =`sqrt(16)/120 = 4/120 = 0.033`

Hence, we see that minimum coefficient of variance is D, hence
product is consistent

Correct Answer is `=>` (D) Product D

Q 2377667586

What is the least value of the standard deviation
of 5 integers, no two of which are equal?
NDA Paper 1 2009

(A)

`sqrt5`

(B)

`2`

(C)

`sqrt2`

(D)

No such least value can be computed

Solution:

Let us consider any five integers be `3, 4, 5, 6` and `7`

its mean = `25/5 = 5`

`therefore SD = sqrt(((5-3)^2+(5-4)^2+(5-5)^2+(5-6)^2+(5-7)^2)/5)`

` = sqrt((4+1+0+1+4)/5)`

` = sqrt2`

Correct Answer is `=>` (C) `sqrt2`

Q 2317767680

Consider the following statements
I. The data, which are collected from the unit or
individual respondents directly for the
purpose of certain study or information are
known as primary data.
II. The data obtained in a census study are
primary data.
Which if the above statements is/are cornet?
NDA Paper 1 2009

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

Both statements are true.

Correct Answer is `=>` (C) Both I and II

Q 2357767684

The average sales and standard deviation of sales
for four months for a company are as follows

During which month are the sales most consistent?
NDA Paper 1 2009

(A)

Month I

(B)

Month 2

(C)

Month 3

(D)

Month 4

Solution:

Month 1, `CV = sigma/x xx100`

` = 2/30xx100 = 6.67`

Month 2 `CV = 3/57xx100 = 5.26`

Month 3 `CV = 4/82 xx100 = 4.68`

Month 4 `CV = 2/28xx100 = 7.14`

Hence, month 3, the sales are most consistent

Correct Answer is `=>` (C) Month 3

Q 2337867782

The marks scored by two students A and B in six
subjects arc given below

Which one of the following statements is most
appropriate?
NDA Paper 1 2009

(A)

The average scores of A and B are same but A is consisten

(B)

The average scores of A ancl B are not same but A is consistent

(C)

The average scores of A and Bare same but B is consistent

(D)

The average scores of A and B are not same but B is consistent

Solution:

Average marks of A

`= (71+56+55+75+54+49)/6 = 360/6 = 60`

`SD = sqrt((121+16+25+225+36+121)/6) = sqrt((544)/6) = 9.52`

Also, average marks of B

`SD = sqrt((16+225+576+25+441+49)/6)`

` = sqrt((1532)/6) = sqrt(255) approx 16`

Now `CV_A = (9.52)/(60)xx100 = 15.87`

and `CV_B = 16/59xx100 = 27.12`
Thus, the average scores of A and B are not same but A is
consistent

Correct Answer is `=>` (B) The average scores of A ancl B are not same but A is consistent

Q 2307867788

In a factory, there are 30 men and 20 women
employees. of the average salary of men is Rs 4050
and the average salary of all the employees is
Rs 3550, then what is the average salary of women'?
NDA Paper 1 2009

(A)

Rs 3800

(B)

Rs 3300

(C)

Rs 3000

(D)

Rs 2800

Solution:

Here, `n = 50, x = 3550, n_1 = 30, x_1 = 4050` and `n_2 = 20`

We know that `nx = n_1x_1+n_2x_2`

` => 50xx3550 = 30xx4050+20x_2`

`=> 177500-121500 = 20x_2`

`=> x_2 = 2800`

Hence, average salary of women = `Rs. 2800`

Correct Answer is `=>` (D) Rs 2800

Q 2387067887

What is the standard deviation of numbers `7, 9,
11, 13` and `15?`
NDA Paper 1 2009

(A)

`2.2`

(B)

`2.4`

(C)

`2.6`

(D)

`2.8`

Solution:

`because barx = (97+9+11+13+15)/5 = 55/5 = 11`

Now

`SD = sqrt(((7-11)^2+(9-11)^2+(11-11)^2+(13-11)^2+(15-11)^2)/5))`

` = sqrt((16+4+0+4+16)/5)`

` = sqrt8 = 2.8` approx

Correct Answer is `=>` (D) `2.8`

Q 2337167982

If the monthly expenditure pattern of a person who
earns a monthly salary of `Rs15000` is represented in
a pie diagram, then the sector angle of an item on
transport expenses measures `15^0`. What is his
monthly expenditure on transport ?
NDA Paper 1 2009

(A)

`Rs 450`

(B)

`Rs 652`

(C)

`Rs 675`

(D)

Cannot be determined

Solution:

Since, monthly salary `=Rs 15000`
and sector angle of expenses `= 15^0`

`therefore ` Required amount = `(15^0)/(360^0)xx15000 = Rs 625`

Correct Answer is `=>` (B) `Rs 652`

Q 2307178088

If `underset(i = 1)overset(n)Sigma(x_i-2) = 110` and `underset(i = 1)overset(n)Sigma(x_i-5) = 20` then what is the mean ?
NDA Paper 1 2009

(A)

`11/2`

(B)

`2/11`

(C)

`17/3`

(D)

`17/9`

Solution:

`because underset(i = 1) overset(n)Sigma(x_i-2) = 110`

`therefore x_1+x_2+................x_n-2n = 110`

`=> x_1+x_2+ ........................x_n = 2n+110` .................(i)

`underset(i = 1)overset(n)Sigma(x_i-5) = 20`

`=> x_1+x_2+..................x_n-5n = 20`

`=> x_1+x_2+..................x_n = 5n+20`.............(ii)

From Eqs. (i) and (ii),

`5n+20 = 2n+110`

` n= 30`

Now Mean = `(x_1+x_2+.................x_n)/n`

` = (5xx30+20)/30 = 170/30 = 17/3`

Correct Answer is `=>` (C) `17/3`

Q 2347478383

What is the arithmetic mean of the series

`text()^nC_1 , text()^nC_2 , text()^nC_3 , .................. text()^nC_n ?`

NDA Paper 1 2008

(A)

`(2^n-1)/n`

(B)

`2^n/(n+1)`

(C)

`(2^n)/n`

(D)

`2^(n+1)/(n+1)`

Solution:

Arithmetic mean of the series

` = (text()^nC_1+text()^nC_2+text()^nC_3 +......................+ text()^nC_n)/n = (2^n-1)/n`

`(because 2^n = text()^nC_0+text()^nC_1+.........................+text()^nC_n)`

`(therefore 2^n -1 = text()^nC_1+text()^nC_2+...................+ text()^nC_n)`

Correct Answer is `=>` (A) `(2^n-1)/n`

Q 2327578481

The average age of 20 students in a class is 15 yr. If
the teacher's age is included, the average increases
by one, then what is the teacher's age?
NDA Paper 1 2008

(A)

30 yr

(B)

21 yr

(C)

42 yr

(D)

36 yr

Solution:

Let the teacher's age is x yr

`therefore 15+1 = (20xx15+x)/(21)`

`=> 16xx21 = 300+x`

`=> x = 336-300 = 36 yr`

Correct Answer is `=>` (D) 36 yr

Q 2317578489

The frequency distribution of a discrete variable `X`
with one missing frequency `f` is given above. If the
arithmetic mean of `X` is `23/8`. what is the value of the

missing frequency?
NDA Paper 1 2008

(A)

`5`

(B)

`6`

(C)

`8`

(D)

`10`

Solution:

Anthmetic mean = `(2xx1+3xx2+3f+4xx5)/(2+3+f+5)`

`=> 23/8 = (28+3f)/(10+f)`

`=> 230+23f = 224+24f`

`therefore f = 6`

Correct Answer is `=>` (B) `6`

Q 2337378282

What is the value of `n` for which the numbers `1, 2,
3, .... , n` have variance `2 '?`
NDA Paper 1 2008

(A)

`4`

(B)

`5`

(C)

`6`

(D)

`8`

Solution:

Mean of the numbers = `(n(n+1)/2)/n`

`barx = (n+1)/2`

`therefore ` Variance = `((1-(n+1)/2)^2+(2-(n+1)/2)^2+(3-(n+1)/2)^2+..............)/n`

` = 2` (given)(`because` Variance `= (underset(i =1)overset(n)Sigma(x- barx)^2/n`)

` 2 = ((1^2+2^2+3^2+......)+n((n+1)/2)^2-2((n+1)/2)(1+2+3+..........))/n`

` => 2n = 1/6n(n+1)(2n+1)+(n(n+1)^2)/4-2((n+1)/2)((n(n+1)/2))`

` => 2n = n(n+1)((2n+1)/6+(n+1)/4-(n+1)/2)`

`=> 2 = (n+1)((4n+2-3n-3)/(12))`

` => 24 = (n+1)(n-1) => n^2-1 = 24`

`=> n^2 = 25`

`n = pm 5`

Correct Answer is `=>` (B) `5`

Q 2337178982

If the three observations are `3, - 6` and `- 6`, then
what is their harmonic mean?
NDA Paper 1 2008

(A)

`0`

(B)

`oo`

(C)

`-1/2`

(D)

`-3`

Solution:

Harmonic mean = `1/3(1/3+1/(-6)+1/(-6))`

`[ because H = n/(1/x_1+1/x_2+......................1/x_n)]`

` = 1/(1/3(1/3-1/3) 1/0 = oo`

Correct Answer is `=>` (B) `oo`

Q 2388801707

Consider the following three methods of collecting
data:
I. Collecting data from government offices
II. Collecting data from public libraries
III. Collecting data by telephonic interview
Select the correct answer using the codes given below
NDA Paper 1 2008

(A)

All the three methods give secondary data

(B)

I and II give secondary and Ill gives primary data

(C)

I and Ill give secondary and II gives primary data

(D)

II and Ill give secondary and I gives primary data

Solution:

Collection data from government offices and public
libraries is secondary and from telephonic interview is primary

Correct Answer is `=>` (B) I and II give secondary and Ill gives primary data

Q 2308801708

NDA Paper 1 2008

Assertion : (A) Data collected in decennial censuses are not statistical data

Reason : (R) Since, no probability is involved in this data collection, it amounts of 100% collection of existing data .

(A)

Both A and R individually true and R is the correct explanation of A

(B)

Both A and R are individually true but R is not the correct explanation of A

(C)

A is true but R is false

(D)

A is false but R is true

Solution:

A and Rare correct, R is the correct explanation of A

Correct Answer is `=>` (A)

Q 2318001800

Consider the following statements
The appropriate number of classes while
constructing a frequency distribution should be
chosen such that
I. the class frequency first increases to a peak
and then declines.
II. the class frequency should duster around the
class mid point.
Which of the statements given above is/are correct?
NDA Paper 1 2008

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

Tile appropriate number of classes while constructing
a frequency distribution should be chosen such that the class
frequency should cluster around the class mid point

Correct Answer is `=>` (B) Only II

Q 2328001801

The populations of four towns A, B, C and D as on
2001 are as follows

What is the most appropriate diagram to present
the above data?
NDA Paper 1 2008

(A)

Pie diagram

(B)

Bar chart

(C)

Cubic chart

(D)

Histogram

Solution:

Required diagram is a bar chart.

Correct Answer is `=>` (B) Bar chart

Q 2348001803

Consider the two series of observations A and B as
follows

If the standard deviation of the Series A is .`sqrt(38)`,
then what is the standard deviation of the Series
B?
NDA Paper 1 2008

(A)

`3.8`

(B)

`sqrt(0.38)`

(C)

`0.38`

(D)

`sqrt(38)`

Solution:

Standard deviation of the Series B'

` = sqrt(1/5(1.9^2+0.8^2+1.5^2+0.6^2+0.2^2)-((1.9+0.8+1.5+0.6+0.2)/5)^2)`

`[ because SD = sqrt((Sigmax_i^2)/n-((Sigmax_i)/n)^2)]`

` = sqrt((6.9)/5)-1 = sqrt(1.38-1)`

` = sqrt(0.38)`

Correct Answer is `=>` (B) `sqrt(0.38)`

Q 2368001805

If `n_1` and `n_2` are the sizes, `G_1` and `G_2` are the
geometric means of two series respectively, which
one of the following expresses the geometric mean
`(G)` of the combined series?

NDA Paper 1 2008

(A)

`logG = (n_1G_1+n_2G_2)/(n_1+n_2)`

(B)

`logG = (n_2logG_1+n_1logG_2)/(n_1+n_2)`

(C)

`G = (n_1logG_1+n_2logG_2)/(n_1+n_2)`

(D)

None of the above

Solution:

Required expression is `logG = (n_2logG_1+n_1logG_2)/(n_1+n_2)`

Correct Answer is `=>` (B) `logG = (n_2logG_1+n_1logG_2)/(n_1+n_2)`

Q 2388001807

Let .`barx` be the mean of n observations `x_1, x_2 , ... , x_n`
If `(a -b)` is added to each observation, what is the
mean of new set of observations?
NDA Paper 1 2008

(A)

`0`

(B)

`barx`

(C)

`barx-(a-b)`

(D)

`barx+(a-b)`

Solution:

S1nce, `barx` is the mean of `n` observations `x_1 ,x_2 , .........................x_n`

`therefore barx = (x_1+x_2+x_3+......................x_n)/n`

Now `(a-b)` is added to each term

`therefore` New mean ` = (x_1+(a-b)+x_2+(a-b)+...............+x_n+(a-b))/n`

` = (x_1+x_2+..................x_n)/n+(n(a-b))/n`

` = barx+(a-b)`

Correct Answer is `=>` (D) `barx+(a-b)`

Q 2318001809

The frequency curve for the distribution of income
in a region is positively skewed as shown in the
figure. Then, for this distribution
NDA Paper 1 2008

(A)

Mean < Mode < Median

(B)

Mode < Median < Mean

(C)

Mode < Mean < Median

(D)

Median < Mean < Mode

Solution:

For given distribution,
Median < Mean < Mode

Correct Answer is `=>` (D) Median < Mean < Mode

Q 2328101901

Students of two schools appeared for a common
test carrying 100 marks. The arithmetic means of
their marks for schools I and II are 82 and 86
respectively. If the number of students of school II
is 1.5 times the number of students of school I,
what is the arithmetic mean of the marks of all the
students of both the schools?
NDA Paper 1 2007

(A)

`84`

(B)

`84.2`

(C)

`84.4`

(D)

Cannot be determined

Solution:

Let the number of students of school I `= x`
`therefore` Number of students of school II `= 1.5x`
Mean of marks of school I = `82`
and mean of marks of school II `= 86`

`therefore ` Combined mean = `(x xx82+1.5x xx86)/(x+1.5x)`

`( because barx_(12) = (barxn_1+barx_2n_2)/(n_1+n_2))`

` = (x(82+129))/(2.5x)`

` = 211/2.5 = 84.4`

Correct Answer is `=>` (C) `84.4`

Q 2348101903

If AM of numbers `x_1, x_2 , ... , x_n` is `mu`, what is the AM
of the numbers which are increased by `1, 2, 3, ... ,n`
respectively?
NDA Paper 1 2007

(A)

`mu+((n+1)/2)`

(B)

`mu`

(C)

`mu+(n(n+1))/2`

(D)

`mu-(n(n+1))/2`

Solution:

Since, AM of numbers `x_1, x_2, x_3, ... , x_n` is `mu`

`therefore mu = (x_1+x_2+ ...........+x_n)/n`

`=> n mu = x_1+x_2 + ..........+x_n`

Sum of the new numbers

` = (x_1+x_2+...........x_n)+(1+2+3+.......n)`

` = mu n+(n(n+1))/2`

`therefore AM = ([mu+((n+1))/2]n)/n`

` = mu+ ((n+1))/2`

Correct Answer is `=>` (A) `mu+((n+1)/2)`

Q 2358101904

In computing a measure of the central tendency for
any set of 51 numbers, which one of the following
measures is well defined but uses only very few of
the numbers of the set?
NDA Paper 1 2007

(A)

Arithmetic mean

(B)

Geometric mean

(C)

Median

(D)

Mode

Solution:

Required measure is mode. (by definition)

Correct Answer is `=>` (D) Mode

Q 2378101906

The data below records the itemvise quarterly
expenditure of a private organization

The data is represented by a pie diagram. What is
the sectorial angle of the sector with largest area?
NDA Paper 1 2007

(A)

`120^0`

(B)

`108^0`

(C)

`100^0`

(D)

`90^0`

Solution:

Largest amount occupies the largest area. Thus, the
salaries occupied largest area.

`therefore` Sectorial angle = `6/20xx360^0 = 108^0`

Correct Answer is `=>` (B) `108^0`

Q 2388101907

NDA Paper 1 2007

Assertion : While constructing the cumulative frequency column of a frequency distribution, it is noticed that these cumulative frequencies are in arithmetic progression. (A) All the class frequencies are equal.

Reason : (R) When all the class frequencies are equal, the cumulative frequencies are in arithmetic progression.

(A)

Both A and R individually true and R is the correct explanation of A

(B)

Both A and R are individually true but R is not the correct explanation of A

(C)

A is true but R is false

(D)

A is false but R is true

Solution:

Both A and R are true and R is the correct explanation
of A.

Correct Answer is `=>` (A)

Q 2318101909

If in a frequency distribution table with 12 classes,
the width of each class is 2.5 and the lowest class
boundary is 6.1, what is the upper class boundary
of the highest class?
NDA Paper 1 2007

(A)

`30.1`

(B)

`27.6`

(C)

`30.6`

(D)

`36.1`

Solution:

Upper class boundary of the highest class
`= 6.1 +(2.5 xx 12) =6.1 +30 =36.1`

Correct Answer is `=>` (D) `36.1`

Q 2328112001

Which one of the following statements is not
correct?
NDA Paper 1 2007

(A)

Median divides distributions into two equal subgroups

(B)

The third quartile is the same as the 75th percentile

(C)

The 5th decile is the same as the 50th percentile

(D)

The 50th decile is the same as the 5th percentile

Solution:

(a) It is true as the median divides the distributions into
two equal subgroups.

(b) Third quartile = `3/4` and 75th percentile = `75/100 = 3/4`

(c) 50th decile `= 5/10 = 1/2` and 50th percentile = `50/100 = 1/2`

(d) 50th decile = `50/10 = 5` and 5th percentile = `5/100 = 1/20`

which is not true

Correct Answer is `=>` (D) The 50th decile is the same as the 5th percentile

Q 2338112002

Frequency curves for the distribution of blood
pressure readings of certain athletes before
exercise (A) and after exercise (B) are plotted
together as shown in the figure above. From the
frequency curves, which one of the following can
be concluded?
NDA Paper 1 2007

(A)

Both distributions are identical

(B)

Both distribu·:ions have the same mean value

(C)

Both distributions have the same mean value but different variances

(D)

Both distributions have the same variance but different mean values

Solution:

From the frequency curves. both the distributions have
the same variance but different mean values.

Correct Answer is `=>` (D) Both distributions have the same variance but different mean values

Q 2358112004

If the slopes of the line of regression of Yon `X` and
of `X` on `Y` are `30` and `60` respectively, then `r(x, y')` is
NDA Paper 1 2007

(A)

`-1`

(B)

`1`

(C)

`1/sqrt3`

(D)

`-1/sqrt3`

Solution:

Here `b_(yx) = tan30^0 = 1/sqrt3`

and `1/b_(xy) = tan60^0 = sqrt3`

`therefore b_(xy)*b_(yx) = 1/3`

`=> r^2 = 1/3 or r = +1/sqrt3`

As `b_(yx)` and `b_(xy)` are positive

Correct Answer is `=>` (C) `1/sqrt3`

Q 2368112005

If you want to measure the intelligence of a group
of students, which one of the following measures
will be more suitable?
NDA Paper 1 2007

(A)

Arithmetic mean

(B)

Mode

(C)

Median

(D)

Geometric rrean

Solution:

If you want to measure the intelligence of a group of
students, then mode will be more suitable

Correct Answer is `=>` (B) Mode

Q 2388112007

If `X` and `Y` are charge into `a+ hU, b + kV`
respectively, then which of the correct relation
between the correlation coefficient `b_(xy)` and `b_(UV)` ?
NDA Paper 1 2007

(A)

`hb_(XY) = kb_(UV)`

(B)

`kb_(XY) = hb_(UV)`

(C)

`b_(XY) = b_(UV)`

(D)

`k^2b_(xy) = h^2b_(UV)`

Solution:

If `X` is changed to `a+ hU` and `Y` to `b + kV`, then

`b_(XY) = |h/k|b_(UV)`

`=> kb_(XY) = hb_(UV)`

Correct Answer is `=>` (B) `kb_(XY) = hb_(UV)`

Q 2713891749

If two regression lines between height `(x)` and weight `(y)` are `4y-15x +410 = 0` and `30x- 2y- 825 = 0`, then what will be the correlation coefficient between height and weight?
NDA Paper 1 2000

(A)

`1/3`

(B)

`1/2`

(C)

`2/3`

(D)

`3/4`

Solution:

If `X` is changed to `a+ hU` and `Y` to `b + kV`, then

`b_(XY) = |h/k|b_(UV)`

`=> kb_(XY) = hb_(UV)`

Correct Answer is `=>` (B) `1/2`